Dimension reduction approach for understanding resource-flow resilience to climate change

The study addresses how to assess and understand resilience of socio-technical, lifeline infrastructure under climate change, focusing on fuel supply chains. Traditional dynamical systems analyses often require detailed dynamics that are unavailable for large, complex, networked systems. Dimension-reduction methods combine network topology with dynamics to yield low-dimensional macroscopic descriptions. The authors extend this approach to resource-flow (conserved) dynamics on multiplex networks and pose the research questions: Can a dimension-reduction framework capture macroscopic flow behavior and stability constraints of a fuel transportation network? How does sea level rise (SLR)–induced coastal flooding affect its ability to meet demand? How does the system transiently respond to production interruptions? A real-world case, the San Francisco Fuel Transportation Network (SFFTN), is used to quantify impacts across climate scenarios and to evaluate demand stability and failure dynamics.

Prior work has explored resilience and robustness in complex networks, generalized modeling for dynamical systems, and the original dimension-reduction framework developed largely in ecological and mutualistic interaction contexts. Supply chain dynamics have been modeled with difference and differential equations, noting the bullwhip effect and challenges due to limited firm-level data. Network-based approaches have been central for large systems, with studies on transport networks, water distribution, and trade. The present work builds on Gao–Barzel–Barabási’s dimension-reduction by adapting it from mutualistic to resource-flow networks that must respect conservation and directional flows, and aligns with generalized modeling that infers stability conditions from structural parameters without detailed functional forms.

System representation: The SFFTN is modeled as a multiplex network with three facility layers: refineries (production), terminals (intermediate storage), and gas stations (consumption). Transportation means define interlayer links: product pipelines between refineries and terminals; road trucking between terminals and gas stations; and a smaller direct trucking link from refineries to gas stations. Nodes located in coastal flood zones are removed per scenario. Data sources include OpenStreetMap, Google Places, U.S. EIA for facility locations, U.S. National Pipeline Mapping for pipelines, and CEC for capacities. Dynamics: Let x_i^q ∈ [0,1] be the stock level at facility i in layer q (q=1 refineries, 2 terminals, 3 gas stations), with stock capacity C^q so stored fuel is C^q x_i^q. Production at refineries is Π(x) with capacity P; demand at gas stations is A(x) with capacity D; interfacility flows have level Ψ(x_i, x_j) with capacity W_{ij}^{qr}. The ODEs encode conservation: production adds to refinery stocks; flows decrease sender and increase receiver stocks; demand removes from gas-station stocks. Dimension reduction: Define layer stock variables y^q as capacity-weighted sums/averages. Under mean-field assumptions of low correlations between capacities and state-dependent level functions, the high-dimensional system reduces to three ODEs: ŷ^1 = p Π(y^1) − s_{12} Ψ(y^1,y^2) − s_{13} Ψ(y^1,y^3) ŷ^2 = s_{12} Ψ(y^1,y^2) − s_{23} Ψ(y^2,y^3) ŷ^3 = d A(y^3) + s_{13} Ψ(y^1,y^3) + s_{23} Ψ(y^2,y^3) with normalized parameters: p=P/C, d=D/C, s_{qr}= (Σ_i Σ_j W_{ij}^{qr})/(N_q C^r), and capacity ratios α_{qr}= (N_q C^q)/(N_r C^r). The normalized total resource is U = y^1 + α_{12} y^2 + α_{23} y^3. Production–demand balance gives Û = p Π(y^1) − α_{12} α_{23} A(y^3). Stable-flow conditions: Setting ŷ^q=0 yields constraints equivalent to a max-flow condition: for stable demand Δ* and production Π*, the feasible macroscopic flows lie on a line in flow space and must satisfy α_{12} α_{23} d Δ* ≤ s_{13} + min(s_{12}, s_{23}). Thus topology (capacities and link availability) directly constrains stable demand independent of specific Π, A, Ψ functional forms. Parameterization: Aggregated natural parameters are estimated from public sources (CEC and others). Representative values/ranges include: N1=5 refineries, N2=29 terminals, N3=3422 gas stations; C^1 ∈ [38.2,57.3] Mgal (per refinery), C^2 ∈ [31,62] Mgal (per terminal, Kinder Morgan estimate), C^3=0.035 Mgal (per gas station). Weekly total interlayer flow capacities: W^{12} (pipelines) ≈ [70,140] Mgal week−1 (via min-cut times single-pipeline capacity), W^{23} (terminals→stations trucks) ≈ [105,245] Mgal week−1 (from ~5000 tanker trips/day with 3–7 kgal trucks), W^{13} (refineries→stations direct trucks) ≈ [21,81] Mgal week−1. Normalized parameters derived include α12, α23, s12, s23, s13, d, and production rate ρ (details in Tables 1–2 of the paper). Functional forms for numerical examples: Π(x) ≈ constant near maximum except close to x=1; A(x) ≈ constant except near x=0; Ψ(x1,x2) increases with sender stock and decreases with receiver stock (e.g., Ψ=x1(1−x2)). SLR scenarios: Four time horizons (2020–2040, 2040–2060, 2060–2080, 2080–2100) under RCP 4.5 and 8.5; four CMIP5 GCMs (CanESM2, MIROC5, CNRM-CM5, HadGEM2-ES); three SLR percentiles (50, 95, 99.9). 3Di hydrodynamic modeling at 50 m resolution generates water-column heights during high sea-level events; nodes with water depth ≥15 cm are deemed failed and removed. Aggregated parameters are recomputed per scenario, including updated facility counts, min-cut for pipelines, and accessible station fractions f_{13}, f_{23} to scale W^{13}, W^{23}. Transient failure experiments: Production interruptions modeled by setting p=0 for duration ΔT ∈ [0.5,3] weeks, with system initialized at stable states spanning feasible U. Metrics: time to demand failure τ (time until y^3 reaches 0) under ΔT=3 weeks; and average supplied demand Q̄ over T=3 weeks, averaged over initial U.

- Dimension-reduction yields an accurate three-ODE macroscopic model that conserves flow and captures layer-level dynamics, reducing >3400 facility equations to 3. - Stable macroscopic flows are constrained by a condition equivalent to max-flow across the multiplex: for a given stable demand, feasible flows lie on a line, and demand is bounded by s_{13} + min(s_{12}, s_{23}), highlighting topology-driven constraints independent of detailed dynamics. - Under SLR, aggregated parameters shift primarily due to facility removal and disconnection: the largest reductions occur in refinery→terminal pipeline capacity s_{12} (≈ −40% for RCP 4.5 and ≈ −75% for RCP 8.5 at worst-case 99.9th percentile scenarios), with similar decreases in refinery→station capacity s_{13}. Effective α_{23} increases as terminals are disproportionately affected; p increases when refineries are lost (holding total production constant). - Demand stability across scenarios: The system sustains original demand through 2040–2060 in all scenarios. Demand failures appear by 2080–2100 under RCP 4.5 and by 2060–2080 and 2080–2100 under RCP 8.5 at high SLR percentiles (95th and 99.9th). Phase-like diagrams show scenario averages moving toward and into the low-demand region as s_{12} and s_{13} decrease. - Scenario probability of meeting target demand (topology-only constraint via stable-flow condition): For RCP 4.5 in 2060–2080 there is only a slight chance to fail at 100% demand; in 2080–2100 up to 60% of parameter realizations cannot sustain 100% demand (99.9th percentile). For RCP 8.5, about 60% fail at 100% demand in 2060–2080 (99.9th), rising to 100% failing for any target demand above 50% of original in 2080–2100. - Transient production interruptions: Time to demand failure τ under a 3-week production stoppage declines as SLR worsens and feasible U shrinks; maximum τ decreases from about 3 weeks in the original network to roughly 1.5 weeks by 2080–2100 (RCP 4.5, 50th percentile), with similar values for RCP 8.5 in 2060–2080 (50th percentile) and for RCP 4.5 in 2060–2080 (95th percentile). When the network cannot supply full demand, τ=0 at the unique feasible U. - Average demand during failure Q̄ over T=3 weeks: For short failures (ΔT<1 week), Q̄ ≈ 1; as ΔT increases, Q̄ decreases approximately linearly with ΔT. Across scenarios, the total reduction in Q̄ is about 10% at ΔT=3 weeks. In failed systems, Q̄ falls below 1 even for ΔT=0.5 week.

The extended dimension-reduction framework captures macroscopic, capacity-conserving dynamics of resource-flow networks and reveals that network topology imposes a max-flow-like constraint on stable demand. This bridges network structure with system function, enabling assessment of demand stability and failure thresholds without detailed micro-level policies or flows. Applied to the SFFTN, the framework shows that coastal flooding under climate change primarily degrades refinery-to-terminal and refinery-to-station capacities and reduces terminal availability, driving the system toward regions where full demand cannot be met in late-century high-SLR scenarios. The transient analysis shows reduced survival times under production interruptions and quantifies performance degradation via average demand, offering actionable insights for contingency planning (e.g., storage strategies, alternative routing, reinforcing pipeline segments, or demand management). The approach complements traditional network metrics by directly relating topology changes to operational outcomes and can inform decision-makers on resilience planning and demand rationing policies.

This work extends the dimension-reduction approach to socio-technical, resource-flow networks and applies it to the San Francisco fuel transportation system under climate-driven coastal flooding. The reduced three-equation model conserves flow and exposes a topology-driven max-flow constraint on stable demand. Scenario analyses indicate the network likely sustains current demand through mid-century but risks demand shortfalls in later decades, especially under high-emission, high-SLR conditions. Transient production failures further reduce system endurance, with survival times shrinking and average demand declining with longer outages. The contributions include: (1) an analytically tractable, system-level model for resource-flow networks; (2) identification of max-flow–equivalent constraints shaping stable macroscopic flows; and (3) quantification of SLR impacts on demand stability and transient failure performance. Future work should incorporate richer heterogeneity and correlations in capacities and policies, validate against detailed operational data, and extend to interdependent systems-of-systems to evaluate cross-infrastructure resilience.

The approximation aggregates facility-level details and assumes low correlations between stock, production, demand, and flow capacities, which may not hold in highly heterogeneous or policy-driven systems. Facility capacities were often assumed equal or estimated from limited public aggregates due to data privacy, potentially obscuring local bottlenecks. Flood impacts were modeled by node removal using a 15 cm water-height threshold and did not alter per-facility capacities other than accessibility-based scaling of aggregate flows, which may underrepresent partial degradation. The chosen functional forms for numerical examples are stylized; while stable-flow constraints are functional-form independent, specific stable stock levels and transients depend on these choices. Results should be viewed as broad estimates contingent on scenario and parameter ranges.